I'm not sure what most of this means, but is the value of $! being set to "no child processes" by one of my waitpid calls, which is interfering with the code in LWP::Protocol::http? Would it help if I localised $! in my reaper sub?

My interpretation of this would be (as you already figured) that $!
is being modified in the signal handler before the interrupted select call
gets a chance to be restarted, i.e. the redo SELECT doesn't execute
because of that very modification of $!.

(Note that because of Perl's deferred (aka safe) signal handling,
the sigreturn() (which is being called at the end of the "real" system/C-level
signal handler) happens immediately, before the Perl signal handler
runs all the waitpid calls. Still, they do run before the
next Perl opcode executes (which means this is presumably beforeif ($!{EINTR} || $!{EAGAIN}) ).

What I find a little surprising is that the ECHILD does occur at
all, because your $Children{$pid} should've been set to zero in
the previous call to the signal handler

where the waitpid did return 14225 (i.e. $res > 0). In other words, you shouldn't be
calling waitpid(14225,...) again thereafter, because the 14225 is no longer supposed to be in the hash... (update: err wait, this is nonsense of course, as you're iterating over the keys, not the values. OTOH, this brings up the question what would happen if you did set the values to the PIDs, too, and then iterate over the values instead (as you seem be to getting that panic when deleting the keys...)

Maybe you could try to figure out why this is — in addition
to trying to localize $! as a workaround, of course.

When putting a smiley right before a closing parenthesis, do you:

Use two parentheses: (Like this: :) )
Use one parenthesis: (Like this: :)
Reverse direction of the smiley: (Like this: (: )
Use angle/square brackets instead of parentheses
Use C-style commenting to set the smiley off from the closing parenthesis
Make the smiley a dunce: (:>
I disapprove of emoticons
Other